Invariants
| Level: | $60$ | $\SL_2$-level: | $10$ | Newform level: | $80$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.48.1.228 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&45\\46&29\end{bmatrix}$, $\begin{bmatrix}23&35\\22&39\end{bmatrix}$, $\begin{bmatrix}49&0\\31&31\end{bmatrix}$, $\begin{bmatrix}49&30\\7&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.24.1.b.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $46080$ |
Jacobian
| Conductor: | $2^{4}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 80.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 41x + 116 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(4:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{8x^{2}y^{6}-1560x^{2}y^{4}z^{2}+94264x^{2}y^{2}z^{4}-1875624x^{2}z^{6}-80xy^{6}z+13392xy^{4}z^{3}-777584xy^{2}z^{5}+15174128xz^{7}-y^{8}+332y^{6}z^{2}-37030y^{4}z^{4}+1771212y^{2}z^{6}-30686529z^{8}}{z^{3}y^{2}(122x^{2}z+xy^{2}-987xz^{2}-15y^{2}z+1996z^{3})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.2.0.a.1 | $4$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 15.24.0-5.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.24.0-5.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.24.0-5.a.1.3 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.96.3-20.b.2.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.96.3-20.d.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.96.3-60.e.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.96.3-60.g.2.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.144.1-20.b.1.5 | $60$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 60.144.5-60.b.2.22 | $60$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.192.5-60.b.2.19 | $60$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.192.5-20.c.1.5 | $60$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.240.5-20.p.1.4 | $60$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 120.96.3-40.b.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3-40.d.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3-120.f.2.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3-120.h.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 300.240.5-100.b.1.1 | $300$ | $5$ | $5$ | $5$ | $?$ | not computed |